Fully extended r-spin topological field theories

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The project is at the interface of pure mathematics – topological field theories (TFTs), higher category theory and algebraic geometry – and theoretical physics. I will use ideas from physics to prove theorems in mathematics.I plan to work on 2-dimensional (2d) fully extended functorial TFTs with tangential structure in the bicategorical and in the (infinity,2)-categorical setting: topological B-twist of Landau-Ginzburg (LG) models described by matrix factorisations and other topologically twisted sigma models that provide certain differential graded (dg) categories.On the way I will apply and develop new results on the LG / (fractional) Calabi-Yau (CY) correspondence, A-infinity categories and homotopy fixed points of circle actions on higher categories categories.The main objective of the project is to find interesting examples of fully extended 2d functorial TFTs on r-spin surfaces by considering different target bicategories and (infinity,2)-categories. I would like to understand the role of fractional CY varieties – related to certain categories of matrix factorisations – as targets of LG models. Another exciting source of examples should come from dg-categories related to topological twists of sigma models in physics.In the bicategorical formulation of fully extended TFTs the proven Cobordism Hypothesis, the computation of the homotopy fixed points of the 2d rotational group, the good understanding of the bicategory of matrix factorisations, and the known conditions on 2-dualisability of dg-categories can be applied.For the second part of the project I need to construct an (infinity,2)-category of matrix factorisations, I will consider algebras in infinity categories, in particular in the dg-nerve of a dg-category to serve as new targets for framed and r-spin TFTs. The project combines state-of-the-art results that became available in the past years to produce new examples of 2d fully extended r-spin TFTs. It is an important task to find such examples in dimension higher than 1 as – although classification conjectures (and theorems in the bicategorical formulation) exist – examples are almost absent in the literature.The result of this project should deliver important insights in topological filed theory and higher category theory and would make a big step towards rigorous mathematical formulation of topologically twisted sigma models in theoretical physics.